I wouldn't say they're "thinking they're something they're not".
The way I look at it after hearing the stories and reading the facts is that they are quite literally born into a body that is different from the body their brain 'thinks' is what they 'should be'. And in my opinion that is not an illness.
To me that sounds like a fundamental definition of a mental illness, people just don't want to apply that label to it because it has historic negative connotations.